Homological and cohomological invariants of electric circuits (Q1778642)

The analysis of electrical networks can be done completely by the use of graph theory. As a consequence, numerical methods were developed for them. The main facts upon which the theory is based consist in: a) expressing all the currents in terms of a current basis (usually the loop currents); b) expressing all the voltages in terms of a voltage basis (usually the node voltages); power conservation law. Adding to them the constitutive relationships, a compatible system of equations providing the global solution is obtained. In the past, another approach due to G. Kron has been given, but in our days the graph theory is largely used. Kron's approach is based on tensor analysis but it cannot give satisfactory answers to some questions (e.g. the power conservation). Here a novel approach, based on the algebraic geometry is presented. It is shown that the geometric structure of the network generates two groups of homologies and cohomologies. In the frame of this approach, the invariance of the input and the output power turns to be a consequence of the geometry of the network (proposition 6 and \(6'\)). Two tables (I and II) comparatively present the main relations for three cases: a) author's theory; b) Kron's theory and c) graph theory. Without any doubt, the present paper provides a nice contribution to the general description of electrical networks. Unfortunately, the chance of the new theory to become usable by electrical engineers is negligible because of the mathematical difficulties for the practitioners to understand the language of the algebraic topology.